The aim of this unit is to teach pupils to: Use accurately the vocabulary, notation and labelling conventions for lines, angles and shapes; distinguish between conventions, facts, definitions and derived properties Identify properties of angles and parallel and perpendicular lines, and use these properties to solve problems Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp 178-183.
A line is the shortest distance between two points. Mathematically, a line only has one dimension, length and no width. We cannot draw a line like this in real life because it would be invisible. The two arrows at either end indicate that the line is infinite. We could not draw an infinitely long line in reality.
Pupils often find the naming of angles difficult, particularly when there is more than one angle at a point. At key stage 3 this confusion is often avoided by using single lower case letters to name angles.
When we discuss lines in geometry, they are assumed to be infinitely long. That means that any two lines in the same plane (that is in the same flat two-dimensional surface) will either intersect at some point or be parallel. This needs to be remembered in the discussion of the pair of parallel lines here. To be parallel, the lines must not intersect no matter how far they are extended.
Pupils should be able to identify parallel and perpendicular lines in 2-D and 3-D shapes and in the environment. For example: rail tracks, double yellow lines, door frame or ruled lines on a page.
Pupils should be able to explain that perpendicular lines intersect at right angles.
Use this activity the identify whether the pairs of lines given are parallel or perpendicular. This activity will also practice the labeling of lines using their end points.
Use this activity to demonstrate that vertically opposite angles are always equal.
Use this activity to demonstrate that the angles on a straight line always add up to 180 °. Hide one of the angles and ask pupils to work out its value. Add another angle to make the problem more difficult.
This should formally summarize the rule that the pupils deduced using the previous interactive slide.
Move the points to change the values of the angles. Show that these will always add up to 360 º. Hide one of the angles, move the points and ask pupils to calculate the size of the missing angle.
This should formally summarize the rule that the pupils deduced using the previous interactive slide.
Point out that that there are two intersecting lines in the second diagram. Click to reveal the solutions.
Ask pupils to give examples of pairs of complementary angles. For example, 32 ° and 58º. Give pupils an acute angle and ask them to calculate the complement to this angle.
Ask pupils to give examples of pairs of supplementary angles. For example, 113 ° and 67º. Give pupils an angle and ask them to calculate the supplement to this angle.
Ask pupils to give any pairs of angles that they think are equal and to explain their choices.
Use this activity to show that when a line crosses a pair of parallel lines eight angles are produced. The four acute angles are equal and the four obtuse angles are equal. The obtuse angle and the acute angle form a pair of supplementary angles. Hide all but one of the angles, move the end points to change the angles and ask pupils to find the value of each hidden angle.
Tell pupils that these are called corresponding angles because they are in the same position on different parallel lines.
Ask pupils to explain how we can calculate the size of angle a using what we have learnt about angle s formed when lines cross parallel lines. If pupils are unsure reveal the hint. When a third parallel line is added we can deduce that a = 29º + 46º = 75º using the equality of alternate angles.
Change the triangle by moving the vertex. Pressing the play button will divide the triangle into three pieces. Pressing play again will rearrange the pieces so that the three vertices come together to form a straight line. Conclude that the angles in a triangle always add up to 180 º. Pupil can replicate this result by taking a triangle cut out of a piece of paper, tearing off each of the corners and rearranging them to make a straight line.
Discuss this proof that angles in a triangle have a sum of 180 º.
Ask pupils to calculate the size of the missing angles before revealing them.
As an alternative to using algebra we could use the following argument: The three angles add up to 180 º, so the two unknown angles must add up to 180º – 88º, that’s 92º. The two angles are the same size, so each must measure half of 92º or 46º.
Drag the vertices of the triangle to show that the exterior angle is equal to the sum of the opposite interior angles. Hide angles by clicking on them and ask pupils to calculate their sizes.
Pupils should be able to understand a proof that the exterior angle is equal to the sum of the two interior opposite angles . Framework reference p183.
Lines and angles
KS3 Mathematics S1 Lines and Angles
Contents S1.1 Labelling lines and angles <ul><li>A </li></ul><ul><li>A </li></ul><ul><li>SS1.4 </li></ul><ul><li>A </li></ul>S1 Lines and angles S1.3 Calculating angles S1.2 Parallel and perpendicular lines S1.4 Calculating angles in triangles andquadrilaterals
Lines In Mathematics, a straight line is defined as having infinite length and no width. Is this possible in real life?
Labelling line segments When a line has end points we say that it has finite length . It is called a line segment . We usually label the end points with capital letters. For example, this line segment has end points A and B. We can call this line ‘line segment AB’. A B
Labelling angles When two lines meet at a point an angle is formed. An angle is a measure of the rotation of one of the line segments relative to the other. We label points using capital letters. A B C Sometimes instead an angle is labelled with a lower case letter. The angle can then be described as ABC or ABC or B.
Contents S1.2 Parallel and perpendicular lines <ul><li>A </li></ul><ul><li>A </li></ul><ul><li>A </li></ul>S1.1 Labelling lines and angles S1.3 Calculating angles S1 Lines and angles S1.4 Calculating angles in triangles andquadrilaterals
Lines in a plane What can you say about these pairs of lines? These lines cross, or intersect . These lines do not intersect. They are parallel .
Lines in a plane A flat two-dimensional surface is called a plane . Any two straight lines in a plane either intersect once … This is called the point of intersection.
Lines in a plane … or they are parallel . We use arrow heads to show that lines are parallel. Parallel lines will never meet. They stay an equal distance apart. Where do you see parallel lines in everyday life? We can say that parallel lines are always equidistant .
Perpendicular lines What is special about the angles at the point of intersection here? a = b = c = d Lines that intersect at right angles are called perpendicular lines. a b c d Each angle is 90 . We show this with a small square in each corner.
Contents S1.3 Calculating angles <ul><li>A </li></ul><ul><li>A </li></ul><ul><li>A </li></ul>S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1 Lines and angles S1.4 Calculating angles in triangles andquadrilaterals
Angles Angles are measured in degrees . A quarter turn measures 90°. It is called a right angle . We label a right angle with a small square. 90°
Angles Angles are measured in degrees . A half turn measures 180°. This is a straight line. 180°
Angles Angles are measured in degrees . A three-quarter turn measures 270°. 270°
Angles Angles are measured in degrees . A full turn measures 360°. 360°
Getting to know angles Use SMILE programs Angle 90 and Angle 360 To get to know angles.
You must learn facts about angles. So you can calculate their size without drawing or measuring. <ul><li>Learn facts about </li></ul><ul><li>Angles between intersecting lines </li></ul><ul><li>Angles on a straight line </li></ul><ul><li>Angles around a point </li></ul>
Angles in a triangle For any triangle, a + b + c = 180 ° The angles in a triangle add up to 180 ° . a b c
Angles in a triangle We can prove that the sum of the angles in a triangle is 180 ° by drawing a line parallel to one of the sides through the opposite vertex. These angles are equal because they are alternate angles. a a b b Call this angle c . c a + b + c = 180 ° because they lie on a straight line. The angles a , b and c in the triangle also add up to 180 ° .
Calculating angles in a triangle Calculate the size of the missing angles in each of the following triangles. 233° 82° 31° 116° 326° 43° 49° 28° a b c d 33° 64° 88° 25°
Calculating angles in a triangle. Calculate the angles shown on this ten ticks worksheet.
Angles in an isosceles triangle In an isosceles triangle , two of the sides are equal. We indicate the equal sides by drawing dashes on them. The two angles at the bottom of the equal sides are called base angles . The two base angles are also equal. If we are told one angle in an isosceles triangle we can work out the other two.
Angles in an isosceles triangle For example, Find the sizes of the other two angles. The two unknown angles are equal so call them both a . We can use the fact that the angles in a triangle add up to 180° to write an equation. 88° + a + a = 180° 88° a a 88° + 2 a = 180° 2 a = 92° a = 46° 46° 46°
Calculating angles in special triangles. Calculate the angles on this ten ticks worksheet.
Interior angles in triangles The angles inside a triangle are called interior angles . The sum of the interior angles of a triangle is 180°. c a b
Exterior angles in triangles f d e When we extend the sides of a polygon outside the shape exterior angles are formed.
Interior and exterior angles in a triangle a b c Any exterior angle in a triangle is equal to the sum of the two opposite interior angles. a = b + c We can prove this by constructing a line parallel to this side. These alternate angles are equal. These corresponding angles are equal. b c
Calculating angles Calculate the size of the lettered angles in each of the following triangles. 82° 31° 64° 34° a b 33° 116° 152° d 25° 127° 131° c 272° 43°
Calculating angles Calculate the size of the lettered angles in this diagram. 56° a 73° b 86° 69° 104° Base angles in the isosceles triangle = (180º – 104º) ÷ 2 = 76º ÷ 2 = 38º 38º 38º Angle a = 180º – 56º – 38º = 86º Angle b = 180º – 73º – 38º = 69º
Sum of the interior angles in a quadrilateral c a b What is the sum of the interior angles in a quadrilateral? We can work this out by dividing the quadrilateral into two triangles. d f e a + b + c = 180° and d + e + f = 180° So, ( a + b + c ) + ( d + e + f ) = 360° The sum of the interior angles in a quadrilateral is 360°.
Sum of interior angles in a polygon We already know that the sum of the interior angles in any triangle is 180°. a + b + c = 180 ° Do you know the sum of the interior angles for any other polygons? a b c a + b + c + d = 360 ° We have just shown that the sum of the interior angles in any quadrilateral is 360°. a b c d
Interior and exterior angles in an equilateral triangle In an equilateral triangle, Every interior angle measures 60°. Every exterior angle measures 120°. The sum of the interior angles is 3 × 60° = 180°. The sum of the exterior angles is 3 × 120° = 360°. 60° 60° 120° 120° 60° 120°
Interior and exterior angles in a square In a square, Every interior angle measures 90°. Every exterior angle measures 90°. The sum of the interior angles is 4 × 90° = 360°. The sum of the exterior angles is 4 × 90° = 360°. 90° 90° 90° 90° 90° 90° 90° 90°